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This extension of the Langmuir isotherm (see Section Multi Component Langmuir) developed in [11] was used to model charge variants of monoclonal antibodies in ion-exchange chromatography. A non-binding salt component \(c_{p,0}\) is added to modulate the ad- and desorption process.

\[\begin{aligned} \frac{\mathrm{d} q_i}{\mathrm{d} t} &= k_{a,i} \exp\left( \frac{k_{\text{act},i}}{T} \right) c_{p,i} q_{\text{max},i} \left( 1 - \sum_{j=1}^{N_{\text{comp}} - 1} \frac{q_j}{q_{\text{max},j}} \right) - k_{d,i} \left( c_{p,0} \right)^{\nu_i} q_i && i = 1, \dots, N_{\text{comp}} - 1 \end{aligned}\]

In this model, the true adsorption rate \(k_{a,i,\text{true}}\) is governed by the Arrhenius law in order to take temperature into account

\[\begin{aligned} k_{a,i,\text{true}} = k_{a,i} \exp\left( \frac{k_{\text{act},i}}{T} \right). \end{aligned}\]

Here, \(k_{a,i}\) is the frequency or pre-exponential factor, \(k_{\text{act},i} = E / R\) is the activation temperature (\(E\) denotes the activation energy and \(R\) the Boltzmann gas constant), and \(T\) is the temperature. The characteristic charge \(\nu\) of the protein is taken into account by the power law.

For more information on model parameters required to define in CADET file format, see Kumar-Langmuir.